Biased Weak Polyform Achievement Games

TL;DR

This paper studies biased weak polyform achievement games on infinite boards, introducing a new priority breaker strategy and determining winners for all small polyforms and all pairs of move counts.

Contribution

It introduces the priority strategy for breakers and completely characterizes winners for all small polyiamonds and polyominoes in the biased game.

Findings

Winners are determined for all pairs (a,b) for polyiamonds up to size four.

A new priority strategy for breakers is introduced.

Winning strategies for the maker are constructed from simpler cases.

Abstract

In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker alternately mark $a,b$ previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the $(a,b)$ game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all $(a,b)$ pairs for polyiamonds and polyominoes up to size four.